Week 9: Statistical Inference

DSAN 5100: Probabilistic Modeling and Statistical Computing
Section 03

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

Open slides in new window →

Schedule

source("../_globals.r")

\[ \DeclareMathOperator*{\argmax}{argmax} \DeclareMathOperator*{\argmin}{argmin} \newcommand{\bigexp}[1]{\exp\mkern-4mu\left[ #1 \right]} \newcommand{\bigexpect}[1]{\mathbb{E}\mkern-4mu \left[ #1 \right]} \newcommand{\convergesAS}{\overset{\text{a.s.}}{\longrightarrow}} \newcommand{\definedas}{\overset{\text{def}}{=}} \newcommand{\definedalign}{\overset{\phantom{\text{def}}}{=}} \newcommand{\eqeventual}{\overset{\mathclap{\text{\small{eventually}}}}{=}} \newcommand{\Err}{\text{Err}} \newcommand{\expect}[1]{\mathbb{E}[#1]} \newcommand{\expectsq}[1]{\mathbb{E}^2[#1]} \newcommand{\fw}[1]{\texttt{#1}} \newcommand{\given}{\mid} \newcommand{\green}[1]{\color{green}{#1}} \newcommand{\heads}{\outcome{heads}} \newcommand{\iid}{\overset{\text{\small{iid}}}{\sim}} \newcommand{\lik}{\mathcal{L}} \newcommand{\loglik}{\ell} \newcommand{\mle}{\textsf{ML}} \newcommand{\nimplies}{\;\not\!\!\!\!\implies} \newcommand{\orange}[1]{\color{orange}{#1}} \newcommand{\outcome}[1]{\textsf{#1}} \newcommand{\param}[1]{{\color{purple} #1}} \newcommand{\pgsamplespace}{\{\green{1},\green{2},\green{3},\purp{4},\purp{5},\purp{6}\}} \newcommand{\prob}[1]{P\left( #1 \right)} \newcommand{\purp}[1]{\color{purple}{#1}} \newcommand{\spacecap}{\; \cap \;} \newcommand{\spacewedge}{\; \wedge \;} \newcommand{\tails}{\outcome{tails}} \newcommand{\Var}[1]{\text{Var}[#1]} \newcommand{\bigVar}[1]{\text{Var}\mkern-4mu \left[ #1 \right]} \]

Today’s Planned Schedule:

	Start	End	Topic	Recording
Lecture	12:30pm	12:40pm	Statistical Inference Overview →
	12:40pm	1:10pm	Asymptotic Laws for Random Samples →
	1:10pm	1:40pm	Maximum Likelihood, GMM Estimation →
Break!	1:40pm	1:50pm
	1:50pm	2:20pm	Bias-Variance Tradeoff Introduction →
Lab	2:20pm	2:40pm	Lab →
	2:40pm	3:00pm	Student Presentation

Statistical Inference

Statistical Inference vs. Statistics

• Moving from understanding probability to getting things done using probability!
• Everything up to this point: understanding the “rules” of stochastic processes
• Now: Using what we know to allow us to draw inferences about populations without having to carry out a census

Samples vs. Populations

• Sample = The data you have
• Population = The thing you want to learn about, by looking at the sample
• Like “success” vs. “failure” in Bernoulli trials, no exact definition of what counts as sample vs. population
• What we call the “sample” and the “population” is vocabulary there to help us know what to do

Inference (Statistical and Otherwise)

What are we doing when we do science?

Science in General

Example: Newton

Remember that the filled-in nodes represent things we can observe, while the non-filled nodes represent things we have to infer from the observable data.

Examples Abound!

Darwinian Evolution

Euclidean Geometry

Zooming in on Euclid

Observations

Source: Emily Pierce

Source: Wikimedia

Interactive visualization from UNC Archaeology

→

Inference

13th-Century Arabic Elements, from Sotheby’s

For an absolutely beautiful visual presentation of Elements see here

Inference in the Diagram

Completing the Cycle

Our Case

If you’re wondering, “what happened to Hypothesis Testing?”, don’t worry, we’ll dive back into that over the next 2 weeks!

Asymptotic Laws for Random Samples

• Law of Large Numbers (LLN)
• Central Limit Theorem (CLT)

Stability out of Randomness

• \(X\) = result of coin flip
• Remembering that \(X\) is a random variable, so it maps outcomes to numbers: \(X(\)\() = 0\), \(X(\)\() = 1\)
• We have no idea what the result of some single coin flip will be, yet we can be sure that the mean of many trials will converge to the expected value of \(0.5\)!

To check that you understand: what value would the mean of many dice rolls converge to?

Interactive Visualization

How Many is “Many”?

library(ggplot2)
library(tibble)
library(dplyr)

Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

n_vals <- c(ceiling(sqrt(10)), 10, ceiling(10*sqrt(10)), 100, ceiling(100*sqrt(10)), 1000, ceiling(1000*sqrt(10)), 10000, ceiling(10000*sqrt(10)), 100000)
heads_data <- c()
total_data <- c()
for (n in n_vals) {
  coin_flips <- rbinom(n, 1, 0.5)
  num_heads <- sum(coin_flips)
  heads_data <- c(heads_data, num_heads)
  num_flipped <- length(coin_flips)
  total_data <- c(total_data, num_flipped)
}
results <- tibble(n = n_vals, heads=heads_data, total=total_data)
results <- results %>% mutate(head_prop = heads / total)
#results
ggplot(results, aes(x=n, y=head_prop)) +
  geom_hline(aes(yintercept=0.5, linetype='dashed'), color=cbPalette[2]) +
  geom_line(aes(color='black'), fill=cbPalette[1], linewidth=g_linewidth, color=cbPalette[1]) +
  geom_point(aes(color='black'), size=g_pointsize*0.9) +
  scale_color_manual("", values=c("black","purple"), labels=c("Mean of n samples","Expected Value")) +
  scale_linetype_manual("", values="dashed", labels="Expected Value") +
  scale_fill_manual("", values=cbPalette[1], labels="95% CI") +
  #scale_linetype_manual('type', values=c('dashed', 'solid'), labels=c("Expected Value", "Sample Mean")) +
  #scale_color_manual('color', values=c(cbPalette[1], 'black'), labels=c('Expected Value', 'Sample Mean')) +
  dsan_theme("full") +
  theme(
      legend.title = element_blank(),
      legend.spacing.y = unit(0, "mm")
  ) +
      labs(
          title = "Estimates of Population Mean for Increasing Sample Sizes",
          x = "n (Sample Size)",
          y = "Sample Mean"
      ) +
  scale_x_log10(breaks = c(10, 100, 1000, 10000, 100000), labels = c("10", "100", "1000", "10000", "100000"))

Warning in geom_line(aes(color = "black"), fill = cbPalette[1], linewidth =
g_linewidth, : Ignoring unknown parameters: `fill`

The Sample Mean

• Like the median we saw earlier (along with lots of other examples), sample mean is just a function of RVs
• Unlike the median, though, it is a function of a vector-valued RV \(\mathbf{X}_N\) (containing \(N\) scalar RVs)!¹
• Let \(\mathbf{X}_N = \{X_1, X_2, \ldots, X_n\}\), where \(X_1\) is first observation, \(X_2\) second observation, and so on. Then:

\[ \overline{X}_N = f(\mathbf{X}_N) = \frac{1}{n}\sum_{i=1}^{n}X_i \]

Weak Law of Large Numbers (WLLN)

• Let \(\mathbf{X}_N = \{X_1, \ldots, X_n\} \iid \mathcal{D}\) be a random sample from a distribution \(\mathcal{D}\) with mean \(\mu\), finite variance. Let \(\overline{X}_N\) denote the sample mean from previous slide. Then

\[ \overline{X}_N \stackrel{p}{\longrightarrow} \mu \]

• \(\mathfrak{X}(N) \stackrel{p}{\longrightarrow} c\) means the Random Variable \(\mathfrak{X}(N)\) (a function of the sample size \(N\)) “converges in probability” to the scalar value \(c\). Formally:

\[ \mathfrak{X}(N) \overset{p}{\longrightarrow} c \iff \forall \varepsilon > 0 \left[ \lim_{N \rightarrow \infty}\Pr\left( | \mathfrak{X}(N) - c | < \varepsilon \right) = 1 \right] \]

Strong Law of Large Numbers (SLLN)

• Same setup as last slide, but

\[ \overline{X}_N \convergesAS \mu \]

• \(\mathfrak{X}(N) \convergesAS c\) means the Random Variable \(\mathfrak{X}(N)\) (a function of the sample size \(N\)) “converges almost surely” to the scalar value \(c\). Formally:

\[ \mathfrak{X}(N) \convergesAS c \iff \Pr\left( \lim_{N \rightarrow \infty} \mathfrak{X}(N) = c \right) = 1 \]

• SLLN \(\implies\) WLLN, but WLLN \(~\nimplies\) SLLN!
• WLLN easy to prove, SLLN very un-easy to prove

Central Limit Theorem

• We’ll never actually reach \(N = \infty\), so we zoom in on how close our sample-based estimate gets to the true value
• Central Limit Theorem says: these “closeness” values are normally distributed!
• LLN guarantees that \(X_N \; \; \eqeventual \; \; \mu\)
• CLT tells us what the gap \((X_N - \mu)\) looks like

Formal CLT

• Sampled observations: \(\mathbf{X}_N = \{X_1, \ldots, X_n\}\)
• \(\expect{X_i} = \mu, \Var{X_i} = \sigma^2 < \infty\)
• \(\overline{X}_N \definedas M_1(\mathbf{X}_N) = \frac{X_1 + \cdots + X_n}{N}\)

\[ \frac{\overline{X}_N - \mu}{\sigma / \sqrt{N}} \overset{\text{asymp}}{\sim} \mathcal{N}(0,1) \]

When is \(\mathcal{N}(0, 1)\) a “good” approximation?

Code

# Prepare data for all plots
max_n <- 10000
num_reps <- 1000
all_rolls <- replicate(
  num_reps,
  sample(1:6, size = max_n, replace = TRUE, prob = rep(1 / 6, 6))
)
gen_clt_plot <- function(n) {
  exp_val <- 3.5
  sigma <- sqrt(35/12)
  denom <- sigma / sqrt(n)
  # Get the slice of all_rolls for this n
  n_rolls <- all_rolls[1:n,]
  sample_means <- colMeans(n_rolls)
  norm_gaps <- (sample_means - exp_val) / denom
  n_df <- tibble(norm_gap=norm_gaps)
  #if (n == 5) {
  #  print(sample_means)
  #  print(n_df)
  #}
  ggplot(n_df, aes(x = norm_gap)) +
  geom_histogram(aes(y = after_stat(density)), binwidth = 1/2) +
  #geom_density() +
  stat_function(fun=dnorm, size=g_linesize) +
  dsan_theme("quarter") +
  labs(
    title = paste0("n = ",n),
    x = "Normalized Sample Gap"
  )
}

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

Q-Q Plots

Code

gen_qq <- function(n) {
  n_rolls <- all_rolls[1:n,]
  sample_means <- colMeans(n_rolls)
  qq_df <- tibble(smean = sample_means)
  qq_plot <- ggplot(qq_df, aes(sample = smean)) +
  stat_qq() + stat_qq_line() +
  dsan_theme("quarter") +
  labs(
    title = paste0("n = ",n)
  )
  return(qq_plot)
}

Overthinking: Estimation

• Maximum Likelihood Estimation (MLE)
• Generalized Method of Moments (GMM) Estimation

Maximum Likelihood Estimation

• Creating a model \(\mathcal{M}\) w/parameters \(\param{\theta}\) means specifying

\[ \mathcal{M} = \Pr(\underbrace{x_1, \ldots, x_n}_{\text{Observed Data}} \mid \underbrace{\param{\theta}}_{\text{Model Parameters}}) \]

• When we view this as a function of \(\param{\theta}\), given the observed data, we call it the likelihood function \(\mathcal{L}_{\mathcal{M}}(\param{\theta} \mid x_1, \ldots, x_n)\)
• Read this as “the likelihood that our model \(\mathcal{M}\), with parameters \(\param{\theta}\), produced the data \(x_1, \ldots, x_n\)”

Probability Models are Generative Models

• A given choice of model parameters \(\param{\theta}\) can be used to generate simulated datapoints!
• Simple example: \(X \sim \text{Bern}(\param{p})\). Just one parameter, \(\param{\theta} = \{\param{p}\}\)
• We observe 10 coin flips: 8 heads, 2 tails. Of all possible Bernoulli distributions (parameterized by \(\param{p}\)), which is most likely to generate this data?

Generative Models

• Given a choice of \(\param{\theta}\) as our golem², we can generate simulated datasets (here 10 for each labeled value of \(\param{p}\)), then compute likelihood as proportion of datasets with 8 heads, 2 tails

• From the plot: (Among these vals) \(\param{p} = 0.8\) is maximum likelihood estimate

library(tibble)
library(ggplot2)
library(dplyr)
set.seed(1948)
obs_heads <- 8
num_flips <- 10
num_reps <- 10
p_vals <- c(0.01, 0.2, 0.4, 0.6, 0.8, 0.99)
obs_matches <- c()
for (i in 1:length(p_vals)) {
  cur_p <- p_vals[i]
  theta_str <- sprintf("%.2f", cur_p)
  sim_data <- replicate(
    num_reps,
    rbinom(num_flips, 1, cur_p)
  )
  #print(sim_data)
  #data_str <- paste0(sim_data, collapse=", ")
  num_heads <- colSums(sim_data)
  #print(num_heads)
  num_matches <- sum(num_heads == obs_heads)
  obs_matches <- c(obs_matches, num_matches)
  #print(num_matches)
  #print(num_heads)
  num_tails <- num_flips - num_heads
  #print(num_tails)
  data_strs <- paste0("[",num_heads," heads, ",num_tails," tails]")
  data_str <- paste0(data_strs, collapse=", ")
  #writeLines(paste0("p = ",theta_str,": ",data_str))
}
#print(obs_matches)
result_df <- tibble(p=as.character(p_vals), num_matches=obs_matches)
result_df <- result_df %>% mutate(prop_matches = obs_matches / num_reps)
ggplot(result_df, aes(x=p, y=prop_matches)) +
  geom_bar(stat = 'identity', fill=cbPalette[1]) +
  dsan_theme("quarter") +
  # theme(
  #   axis.title.y = element_text(size = 12)
  # ) +
  labs(
    title = "Likelihood of data (8 heads, 2 tails) given p",
    y = "Proportion of times (8,2) generated"
  )

Simulations \(\rightarrow\) Math

Prev example was overkill: we can solve for optimal \(\param{p}\) value…

\[ \begin{align*} p^* &\overset{\phantom{x_i\text{ indep}}}{=} \argmax_{\param{p}} \mathcal{L}(x_1, \ldots, x_n \mid \param{p}) \\ &\overset{x_i\text{ indep}}{=} \argmax_{\param{p}} \mathcal{L}(x_1 \mid \param{p})\mathcal{L}(x_2 \mid \param{p}) \cdots \mathcal{L}(x_n \mid \param{p}) \end{align*} \]

1. What are the individual \(\mathcal{L}(x_i \mid \param{p})\) terms?
2. How do we maximize the product \(\mathcal{L}(x_1 \mid \param{p}) \cdots \mathcal{L}(x_n \mid \param{p})\)?

Time for some Math Magic…

Math Magic 1

1. What are the individual \(P(x_i \mid \param{p})\) terms?

\(X \sim \text{Bern}(\param{p})\), so

\[ \begin{align*} \Pr(X = x_i \mid \param{p}) &= \begin{cases}1 - \param{p} & x_i = 0 \\ \param{p} & x_i = 1\end{cases} \; \leftarrow \genfrac{}{}{0pt}{}{\text{ Non-differentiable}}{😭} \\ &\overset{\text{math}}{\underset{\text{magic}}{=}} (1-\param{p})^{1-x_i}\param{p}^{x_i} \; \leftarrow \text{ Differentiable! 😲} \end{align*} \]

Why do we need it to be differentiable? Stay tuned…

Math Magic 2

2. How do we maximize the product?

\[ p^* = \argmax_{\param{p}} f(\param{p}) \implies f'(p^*) = 0 \]

To maximize likelihood, we need to find its derivative³, set equal to 0, and solve:

\[ \begin{align*} &\frac{d}{d\param{p}}\lik(x \mid \param{p}) = 0 \iff \\ &\frac{d}{d\param{p}}\left[\lik(x_1 \mid \param{p})\lik(x_2 \mid \param{p})\cdots \lik(x_n \mid \param{p})\right] = 0 \end{align*} \]

Obstacle: Products vs. Sums

• Finding \(\frac{d}{d\param{p}}\left[ \lik(x_1)\lik(x_2)\cdots \lik(x_n)\right]\) is a doozy, even with just \(n = 2\) datapoints:

\[ \begin{align*} &\frac{d}{d\param{p}}\left[ \lik(x_1)\lik(x_2) \right] = \left( \frac{d}{d\param{p}}\lik(x_1)\right) \cdot \lik(x_2) + \lik(x_1)\cdot \left( \frac{d}{d\param{p}}\lik(x_2) \right) \\ &= (1-\param{p})^{-x_1}\param{p}^{x_1-1}(x_1-\param{p})\cdot (1-\param{p})^{1-x_2}\param{p}^{x_2} \\ &+ (1-\param{p})^{1-x_1}\param{p}^{x_1} \cdot (1-\param{p})^{-x_2}\param{p}^{x_2-1}(x_2 - \param{p}) %&= \frac{d}{d\theta}\left[ (1-p)^{1-x_1}p^{x_1}(1-p)^{1-x_2}p^{x_2} \right] \end{align*} \]

• Complicating factor: \(\lik(x_i)\) terms are all multiplied together, forcing us to use product rule: \(\frac{d}{d\param{p}}\left[\lik(x_1)\lik(x_2)\right] = \left(\frac{d}{d\param{p}}\lik(x_1)\right)\cdot \lik(x_2) + \lik(x_1)\cdot \left(\frac{d}{d\param{p}}\lik(x_2)\right)\)
• If we had terms that were added rather than multiplied, we’d have a much easier time: \(\frac{d}{d\param{p}}\left[ \lik(x_1) + \lik(x_2)\right] = \frac{d}{d\param{p}}\lik(x_1) + \frac{d}{d\param{p}} \lik(x_2)\)⁴
• So, what math operation do we know that turns multiplications into additions?

Math Magic 3: Log-Likelihood

\[ \log(a\cdot b) = \log(a) + \log(b) \]

Bingo! So, can we maximize \(\loglik(x_i) = \log(\mathcal{L}(x_i))\) rather than \(\mathcal{L}(x_i)\)? Bingo again! Because logarithms are monotonic,

\[ x^* = \argmax_x \left[ \log\left(f(x)\right) \right] \iff x^* = \argmax_x \left[ f(x) \right] \]

So, we can just solve

\[ p^* = \argmax_{\param{p}} \left[ \ell(x_1, \ldots, x_n)\right] \]

Simplifying

Our problem simplifies to figuring out

\[ \begin{align*} \frac{d}{d\param{p}}\left[ \log \left( \lik(x_1)\cdots \lik(x_n) \right) \right] &= \frac{d}{d\param{p}}\left[ \log \lik(x_1) + \log\lik(x_2) + \cdots + \log\lik(x_n) \right] \\ &= \frac{d}{d\param{p}}\left[ \ell(x_1) + \ell(x_2) + \cdots + \ell(x_n) \right] = \frac{d}{d\param{p}}\left[ \sum_{i=1}^n \ell(x_i)\right] \end{align*} \]

But since the derivative is an additive operator, \(\frac{d}{d\param{p}}\left[ \sum_{i=1}^n \ell(x_i) \right] = \sum_{i=1}^n \frac{d}{d\param{p}}\left[ \ell(x_i) \right]\), so we just have to compute \(\frac{d}{d\param{p}}\ell(x_i)\)! No product rule required (we still need chain rule):

\[ \begin{align*} \frac{d}{d\param{p}}\left[ \ell(x_i) \right] &= \frac{d}{d\param{p}}\left[ \log((1-\param{p})^{1-x_i}\param{p}^{x_i}) \right] = \frac{d}{d\param{p}}\left[(1-x_i)\log(1-\param{p}) + x_i\log(\param{p})\right] \\ &= (1-x_i)\frac{d}{d\param{p}}\log(1-\param{p}) + x_i\frac{d}{d\param{p}}\log(\param{p}) = -\frac{1-x_i}{1-\param{p}} + \frac{x_i}{\param{p}} \\ &= \frac{\param{p} - x_i}{(\param{p}-1)\param{p}} \end{align*} \]

Maximizing

Now that we know \(\frac{d}{d\param{p}}\ell(x_i)\), we set our log-likelihood equation equal to zero to find the likelihood-maximizing \(\param{p}\) value, \(p^*\):

\[ \begin{align*} &\sum_{i=1}^n\frac{d}{d\param{p}}\ell(x_i) = 0 \iff \sum_{i=1}^n \frac{p^* - x_i}{(p^*-1)p^*} = 0 \\ &\iff -\frac{1}{(p^*-1)p^*}\sum_{i=1}^nx_i - np^* = 0 \\ &\iff \sum_{i=1}^nx_i = np^* \iff \boxed{p^* = \frac{\sum_{i=1}^nx_i}{n}} \end{align*} \]

MLE Intuition

\[ p^* = \frac{\sum_{i=1}^nx_i}{n} = \frac{\sum_{i=1}^n \mathbf{1}[x_i = 1]}{n} \genfrac{}{}{0pt}{}{\leftarrow \text{\# Heads}}{\leftarrow \text{\# Flips }} \]

• MLE almost always matches intuition! Example: given data \(x_1, \ldots, x_n\), what Normal distribution best fits this data?
• Same as asking: what parameter settings for \(\mathcal{N}(\param{\mu}, \param{\sigma^2})\) are most likely to produce \(x_1, \ldots, x_n\)? The answer:

\[ \mu^* = \frac{\sum_{i=1}^n x_i}{n} \; \; \; \sigma^2_* = \frac{\sum_{i=1}^n (x_i-\mu^*)^2}{n} \]

The Dark Side of MLE

• Sometimes steers us in the wrong direction!
• Consider values from previous slide, as estimators for population \(\mu\) and \(\sigma^2\): \(\mu^*\) unbiased if \(\expect{\mu^*} = \mu\):

\[ \begin{align*} \expect{\mu^*} &= \bigexpect{\frac{\sum_{i=1}^nx_i}{n}} = \frac{1}{n}\sum_{i=1}^n\expect{x_i} \\ &= \frac{1}{n}\sum_{i=1}^n\mu = \frac{n\mu}{n} = \mu \; ✅ \end{align*} \]

• So far so good. How about \(\sigma^2_*\)?

MLE as Biased Estimator

• Before we think about \(\expect{\sigma^2_*}\), let’s rewrite \(\sigma^2_*\):

\[ \begin{align*} \sigma^2_* &= \frac{1}{n}\sum_{i=1}^n(x_i - \mu^*)^2 = \frac{1}{n}\sum_{i=1}^n \left( x_i^2 - 2 \mu^* x_i + (\mu^*)^2 \right) \\ &= \frac{1}{n}\sum_{i=1}^nx_i^2 - 2\mu^*\underbrace{\frac{\sum_{i=1}^nx_i}{n}}_{\mu^*} + (\mu^*)^2 = \frac{1}{n}\sum_{i=1}^nx_i^2 - (\mu^*)^2 \\ &= \frac{1}{n}\sum_{i=1}^nx_i^2 - (\mu^*)^2 \end{align*} \]

• Now we’re ready to compute \(\expect{\sigma^2_*}\)!

Computing \(\expect{\sigma^2_*}\)

\[ \begin{align*} \expect{\sigma^2_*} &= \bigexpect{\frac{1}{n}\sum_{i=1}^nx_i^2 - (\mu^*)^2} = \frac{1}{n}\sum_{i=1}^n \expect{x_i^2} - \expect{(\mu^*)^2} \end{align*} \]

• What do we know about \(\expect{x_i^2}\)? Remember the (alternate) definition of variance: \(\Var{X} = \expect{X^2} - \left(\expect{X}\right)^2\). Then

\[ \expect{X^2} = \Var{X} + \left(\expect{X}\right)^2 \]

So let’s plug in the right side when we see \(\expect{X^2}\) or \(\expect{(\mu^*)^2}\):

\[ \begin{align*} \frac{1}{n}\sum_{i=1}^n \expect{x_i^2} - \expect{(\mu^*)^2} &= \frac{1}{n}\sum_{i=1}^n\left(\Var{X} + \left(\expect{X}\right)^2\right) - \expect{(\mu^*)^2} \\ &= (\sigma^2 + \mu^2) - \left(\Var{\mu^*} + \left(\expect{\mu^*}\right)^2\right) \end{align*} \]

Almost There!

We know that \(\expect{\mu^*} = \mu\), but what is \(\Var{\mu^*}\)? Remember that \(\Var{aX} = a^2\Var{X}\)!

\[ \Var{\mu^*} = \bigVar{\frac{1}{n}\sum_{i=1}^nx_i} = \frac{1}{n^2}\sum_{i=1}^n\Var{x_i} = \frac{n\sigma^2}{n^2} = \frac{\sigma^2}{n} \]

And we plug back in to get:

\[ \begin{align*} \expect{\sigma^2_*} &= \sigma^2 + \mu^2 - \Var{\mu^*} - \left(\expect{\mu^*}\right)^2 \\ &= \sigma^2 + \mu^2 - \frac{\sigma^2}{n} - \mu^2 = \sigma^2 - \frac{\sigma^2}{n} \\ &= \frac{n\sigma^2 - \sigma^2}{n} = \frac{\sigma^2(n-1)}{n} \\ &= \color{red}{\left(\frac{n-1}{n}\right)\sigma^2} \neq \sigma^2 \; 💀 \end{align*} \]

Why Does This Happen?: Handwaving

• Long story short, we underpredict the population variance because we already used some of the data to compute \(\mu^*\)!
• This is where the heuristic of degrees of freedom comes in:
  • When we construct an estimate \(e\), \(df(e) = n - k_e\)
  • \(k_e =\) number of other estimates used to calculate \(e\)!

Handwavy Intuition

• Consider \(X_1, X_2 \sim \mathcal{N}(0,1)\): \(\mu = 0\), \(\sigma^2 = 1\).

\[ \expect{\mu^*} = \bigexpect{\frac{X_1 + X_2}{2}} = \frac{1}{2}\left(\expect{X_1} + \expect{X_2}\right) = 0 = \mu \; ✅ \]

\[ \begin{align*} \expect{\sigma^2_*} &= \frac{1}{2}\bigexpect{(X_1 - \mu^*)^2 + (X_2 - \mu^*)^2} \\ &= \frac{1}{2}\bigexpect{\left(X_1^2 + \mu^2_* -2\mu^*X_1\right) + \left(X_2^2 + \mu^2_* - 2X_2\mu^*\right)} \\ &= \frac{1}{2}\expect{X_1^2 + X_2^2 + 2\mu_*^2 - 2\mu^*X_1 - 2\mu^*X_2} \\ &= \frac{1}{2}\left(\expect{X_1^2} + \expect{X_2^2} + 2\expect{\mu_*^2} - 2\expect{\mu^*X_1} - 2\expect{\mu^*X_2}\right) \\ &\implies \boxed{\expect{\sigma^2_*} = \frac{1}{2}\sigma^2} \end{align*} \]

• We’re off by \(\frac{1}{2}\)! What to do?

Handwavy Solution

• We can account for degrees of freedom, correcting the MLE by a factor of \(\frac{n}{df(e^*)}\)!
  • \(e^\circledast = \frac{n}{df(e^*)}e^*\)
• Ex: Since \(\expect{\sigma_*^2} = \frac{n-1}{n}\sigma^2\), we can instead use \(\sigma^2_\circledast = \frac{n}{n-1}\sigma^2_*\). This gives us:

\[ \expect{\sigma^2_\circledast} = \bigexpect{\frac{n}{n-1}\sigma^2_*} = \frac{n}{n-1}\frac{n-1}{n}\sigma^2 = \color{green}{\sigma^2} \; ✅ \]

21st-Century Solution

• Be Bayesian, use priors on parameters (creating hyperparameters)!
• Pretend we know \(\sigma^2\), but want to find the “best” value of \(\mu\):

\[ \begin{array}{rlccc} X_1, X_2 \overset{iid}{\sim} \mathcal{N}( &\hspace{-5mm}\mu\hspace{0.5mm}, &\hspace{-8mm}\overbrace{\sigma^2}^{\large\text{known}}\hspace{-2mm}) & & \\ &\hspace{-4mm}\downarrow & ~ &\hspace{-10mm}{\small\text{estimate}} & \hspace{-6mm} & \hspace{-8mm}{\small\text{uncertainty}} \\[-5mm] &\hspace{-5mm}\mu &\hspace{-5mm}\sim \mathcal{N}&\hspace{-7mm}(\overbrace{m}&\hspace{-12mm}, &\hspace{-16mm}\overbrace{s}) \end{array} \]

Single Datapoint

• Let’s consider the estimate of \(\mu\) from a single datapoint \(X_i\). MLE just gives us \(\mu^* = X_i\). How about MAP estimate?

\[ \lik\left(X_i, \mu, m, s\right) \overset{\text{factors}}{\underset{\text{into}}{=}} P(X_i \mid \mu)P(\mu \mid m, s)P(m, s) \]

• Remembering the pdf of the Normal distribution, we have:

\[ \lik\left(X_i, \mu, m, s\right) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{(X_i-\mu)^2}{2\sigma^2}\right]\frac{1}{s\sqrt{2\pi}}\exp\left[-\frac{(\mu - m)^2}{2s^2}\right] \]

• Then, remembering that we can maximize the log-likelihood rather than the likelihood:

\[ \ell(X_i, \mu, m, s) = \log\left[\frac{1}{\sigma\sqrt{2\pi}}\right] - \frac{(X_i-\mu)^2}{2\sigma^2} + \log\left[\frac{1}{s\sqrt{2\pi}}\right] - \frac{(\mu - m)^2}{2s^2} \]

Taking the Derivative

• Taking the derivative gives us:

\[ \begin{align*} \frac{\partial\ell}{\partial \mu} &= \frac{\partial}{\partial\mu}\left[ {\color{red}\cancel{\color{black}\log\left[\frac{1}{\sigma\sqrt{2\pi}}\right]}} - \frac{(X_i-\mu)^2}{2\sigma^2} + {\color{red}\cancel{\color{black}\log\left[\frac{1}{s\sqrt{2\pi}}\right]}} - \frac{(\mu - m)^2}{2s^2}\right] \\ &= - \frac{1}{2\sigma^2}\cdot\frac{\partial}{\partial\mu}\left[{\color{red}\cancel{\color{black}X_i^2}} + \mu^2 - 2X_i\mu\right] - \frac{1}{2s^2}\cdot\frac{\partial}{\partial\mu}\left[\mu^2 + {\color{red}\cancel{\color{black}m^2}} - 2\mu m\right] \\ &= -\frac{1}{2\sigma^2}\cdot (2\mu -2X_i) - \frac{1}{2s^2}\cdot (2\mu - 2m) = \frac{X_i-\mu}{\sigma^2} + \frac{m - \mu}{s^2} \end{align*} \]

• And we set equal to zero and solve to obtain the MAP estimate:

\[ \begin{align*} &\frac{X_i - \mu^*}{\sigma^2} + \frac{m - \mu^*}{s^2} = 0 \iff \frac{\mu^*}{\sigma^2} + \frac{\mu^*}{s^2} = \frac{X_i}{\sigma^2} + \frac{m}{s^2} \iff \\ &\frac{s^2\mu^* + \sigma^2\mu^*}{\sigma^2s^2} = \frac{s^2X_i + \sigma^2m}{\sigma^2s^2} \iff \mu^*(s^2+\sigma^2) = s^2X_i + \sigma^2m \\ &\iff \boxed{\mu^* = \left(\frac{s^2}{s^2 + \sigma^2}\right)X_i + \left(\frac{\sigma^2}{s^2 + \sigma^2}\right)m} \end{align*} \]

The Takeaway

• Bayesian approach allows new evidence to be weighed against existing evidence, with statistically principled way to derive these weights:

\[ \begin{array}{ccccc} P_{\text{post}}(\mathcal{H}) &\hspace{-6mm}\propto &\hspace{-6mm} P(X \mid \mathcal{H}) &\hspace{-6mm} \times &\hspace{-6mm} P_{\text{pre}}(\mathcal{H}) \\ \text{Posterior} &\hspace{-6mm}\propto &\hspace{-6mm}\text{Evidence} &\hspace{-6mm} \times &\hspace{-6mm} \text{Prior} \end{array} \]

Generalized Method of Moments (GMM) Estimation

• Recall that the \(k\)th moment of an RV \(X\) is \(\mu_k = \expect{X^k}\)

• e.g., \(\mu_1 = \expect{X}\), \(\mu_2 = \Var{X} + \expect{X}^2\)

• Also recall (I rambled on about) how the MGF contains all information about a distribution. This means we can estimate distributions from data:

• Define \(k\)th sample moment of \(\mathbf{X}_N\) to be \(\widehat{\mu}_k = \frac{1}{N}\sum_{i=1}^nX_i^k\). Then the system of equations:

  \[ \begin{align*} \mu_1(\param{\theta}) &= \widehat{\mu}_1 \\ \mu_2(\param{\theta}) &= \widehat{\mu}_2 \\ &\vdots \\ \mu_N(\param{\theta}) &= \widehat{\mu}_N \end{align*} \]

  Gives us a system of equations, allowing us to solve for parameters of our distribution!

The Bias-Variance Tradeoff

There’s no such thing as a free lunch.

Life / Conservative Economists

But modern Machine Learning basically gets us rly close to a free lunch

Jeff

Intuition

	Low Variance	High Variance
Low Bias
High Bias

Figure adapted from Fortmann-Roe (2012), “Understanding the Bias-Variance Tradeoff”

Math

• We estimate “true” DGP \(Y = f(X)\) with model \(\widehat{f}(X)\)⁵, and then we use \(\widehat{f}\) to predict the value of \(Y\) for a point \(x_0\).
• What is our expected error at this point, \(\Err(x_0)\)?

\[ \begin{align*} \Err(x_0) &= \bigexpect{\left.(Y − \widehat{f}(x_0))^2 \right| X = x_0} \\ &= \sigma^2_{\varepsilon} + \left( \bigexpect{\widehat{f}(x_0)} − f(x_0) \right)^2 + \mathbb{E}\left[\widehat{f}(x_0) − \bigexpect{\widehat{f}(x_0)}\right]^2 \\ &= \sigma^2_{\varepsilon} + \left( \text{Bias}(\widehat{f}(x_0)\right)^2 + \bigVar{\widehat{f}(x_0)} \\ &= \text{Irreducible Error} + \text{Bias}^2 + \text{Variance}. \end{align*} \]

In Practice

Figure from Tharwat (2019)

Appendix 1: Derivation of \(\mathbb{E}[\mu_*^2]\)

\[ \begin{align*} \expect{\mu_*^2} &= \bigexpect{\left(\frac{X_1 + X_2}{2}\right)^2} = \frac{1}{4}\expect{(X_1+X_2)^2} \\ &= \frac{1}{4}\expect{X_1^2 + X_2^2 + 2X_1X_2} = \frac{1}{4}\left(\expect{X_1^2} + \expect{X_2^2} + 2\expect{X_1X_2}\right) \\ &= \frac{1}{4}\left(\Var{X_1} + \expect{X_1}^2 + \Var{X_2} + \expect{X_2}^2 + 2\expect{X_1X_2}\right) \\ &= \frac{1}{4}\left(2\sigma^2 + 2\mu^2 + 2\expect{X_1X_2}\right) \\ &= \frac{1}{2}\left(\sigma^2 + \mu^2 + \expect{X_1X_2}\right) \overset{iid}{=} \frac{1}{2}\left(\sigma^2 + \mu^2 + \mu^2\right) \\ &\implies \boxed{\expect{\mu^2_*} = \mu^2 + \frac{\sigma^2}{2}} \; \; \left(\therefore \; \expect{\mu_*^2} \neq \mu^2 \right) \end{align*} \]

Appendix 2: Derivation of \(\mathbb{E}[\mu^*X_i]\)

\[ \begin{align*} \expect{\mu^*X_1} &= \bigexpect{\left(\frac{X_1 + X_2}{2}\right)X_1} = \frac{1}{2}\expect{X_1^2 + X_1X_2} \\ &= \frac{1}{2}\expect{X_1^2} + \frac{1}{2}\expect{X_1X_2} = \frac{1}{2}\left(\sigma^2 + \mu^2\right) + \frac{1}{2}\mu^2 \\ &\implies \expect{\mu^*X_1} = \mu^2 + \frac{\sigma^2}{2} \end{align*} \]

And since \(X_1\) was chosen without loss of generality,

\[ \boxed{\expect{\mu^*X_i} = \mu^2 + \frac{\sigma^2}{2}} \]

Appendix 3: Derivation of \(\mathbb{E}[\sigma^2_*]\)

\[ \begin{align*} \expect{\sigma^2_*} &= \frac{1}{2}\bigexpect{(X_1 - \mu^*)^2 + (X_2 - \mu^*)^2} \\ &= \frac{1}{2}\bigexpect{\left(X_1^2 + \mu^2_* -2\mu^*X_1\right) + \left(X_2^2 + \mu^2_* - 2X_2\mu^*\right)} \\ &= \frac{1}{2}\expect{X_1^2 + X_2^2 + 2\mu_*^2 - 2\mu^*X_1 - 2\mu^*X_2} \\ &= \frac{1}{2}\left(\expect{X_1^2} + \expect{X_2^2} + 2\expect{\mu_*^2} - 2\expect{\mu^*X_1} - 2\expect{\mu^*X_2}\right) \\ &= \frac{1}{2}\left( 2\sigma^2 + 2\mu^2 + 2\left(\mu^2 + \frac{\sigma^2}{2}\right) - 2\left(\mu^2 + s/2\right) - 2\left(\mu^2 + s/2\right) \right) \\ &= \sigma^2 + \mu^2 + \mu^2 + \frac{\sigma^2}{2} - \mu^2 - \frac{\sigma^2}{2} - \mu^2 - \frac{\sigma^2}{2} = \sigma^2 - \frac{\sigma^2}{2} \\ &\implies \boxed{\expect{\sigma^2_*} = \frac{1}{2}\sigma^2} \end{align*} \]

References

Collins, Harry M., and Trevor Pinch. 1993. The Golem: What You Should Know About Science. Cambridge University Press.

Hastie, Trevor, Robert Tibshirani, and Jerome Friedman. 2013. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Science & Business Media.

McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and STAN. CRC Press.

Tharwat, Alaa. 2019. “Parameter Investigation of Support Vector Machine Classifier with Kernel Functions.” Knowledge and Information Systems 61 (3): 1269–1302. https://doi.org/10.1007/s10115-019-01335-4.

Footnotes

1. Note that I’m trying really hard to differentiate between (capital-letter) \(N\) as cardinality of observations, then (lowercase) \(1, 2, \ldots, n\) for ordinal labels on each individual observation. This will spare you many headaches in life!

2. A big rock boi from the Talmud (and Pokémon) who follows orders but has no common sense/nuance (like a computer). See Collins and Pinch (1993) and McElreath (2020)

3. And that’s why we used math magic to make \(\Pr(x_i \mid \param{p})\) differentiable, in the previous slide!

4. We achieve this simplification because the derivative operator is additive: \(\frac{d}{dx}\left[ f(x) + g(x) \right] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)\)

5. It’s even more complicated, since we don’t even know whether the features \(X\) we’ve chosen are actually the features in the world that causally affect \(Y\), but that’s for later classes… Or see Hastie, Tibshirani, and Friedman (2013)!